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Since the integrand
n
F=H- S
Pixcfw_
(12.73)
/= i
depends on x, u, and t, there are n + m dependent variables (x and u) and
hence the Euler-Lagrange equations become
?-7,(T)
dxt
=0
'
'-!A.-
< ( ) _ o. j - u
duj
02-74)
dt \dxj
(12 .75)
dt \
COURSE: OPTIMIZATION TECHNIQUE AAOC C222
ASSIGNMENT - 5
Date of Assignment: 06/11/09
Date of Submission: 13/11/09 (Common Hour)
Maximum marks: 5
Q1. Solve the given assignment model (refer to Table 1) by the Hungarian Method and determine
the associated o
INTEGER LINEAR PROGRAMMING
There are many LP problems in which the decision
variables will take only integer values. If all the
decision variables will only take integer values it is
called a pure integer LPP; otherwise the problem is
called a mixed integ
CLASSICAL OPTIMIZATION THEORY
Quadratic forms
Let
x1
x
2
X .
.
xn
be a n-vector.
Let A = ( aij) be a nn symmetric matrix.
We define the kth order principal minor as
the kk determinant
a11 a12 . a1k
a21 a22 . a2 k
.
.
ak 1 ak 2 . akk
Then the q
QUADRATIC
PROGRAMMING
Quadratic Programming
A quadratic programming problem is a non-linear
programming problem of the form
Maximize
Subject to
T
z c X X DX
A X b , X 0
Here
x1
b1
x
b
2 2
X . , b . , c c1 c2 . . . cn
.
.
xn
bm
a11 a12
Dual simplex method for
solving the primal
In this lecture we describe the
important Dual Simplex method
and illustrate the method by doing
one or two problems.
Dual Simplex Method
Suppose a basic solution satisfies the optimality
conditions but not feasi
GAME THEORY
Life is full of conflict and competition.
Numerical examples involving adversaries in
conflict include parlor games, military battles,
political campaigns, advertising and
marketing campaigns by competing business
firms and so forth. A basic f
Iterative
computations of the
Transportation
algorithm
Iterative computations of the Transportation algorithm
After determining the starting BFS by any one of the
three methods discussed earlier, we use the following
algorithm to determine the optimum sol
Sensitivity Analysis
The optimal solution of a LPP is based on the
conditions that prevailed at the time the LP model
was formulated and solved. In the real world, the
decision environment rarely remains static and it
is essential to determine how the opt
CPM and
PERT
CPM and PERT
CPM (Critical Path Method) and PERT
(Program Evaluation and Review Technique)
are network based methods designed to assist
in the planning, scheduling, and control of
projects. A project is a collection of
interrelated activities
The Assignment Model
" The best person for job" is an apt description of
the assignment model.
The general assignment model with n workers
and n jobs is presented below:
Jobs
1 2 .
n
1 c11 c12
c1n
Workers 2 c21 c22
c2n
n
cn1 cn2
cnn
The element cij is
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
I Semester 2007-08 AAOC C222 (Optimization)
Assignment 3
*
Note: Submit handwritten solutions of the assignment to your instructor on regular class
(i.e. 31/10/2007 in case of Monday series or 30/10/2007 i
COURSE: OPTIMIZATION TECHNIQUE AAOC C222
ASSIGNMENT-1
Date of Assignment: 21/08/09 Date of Submission 28/8/09 (Common Hour)
Maximum marks: 5
Q1. A call center has the following minimal daily requirement for personnel as shown in Table
below. Consider peri
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
Semester-I, 2007-08 AAOC C222 (Optimization)
Assignment 4
*
Note: Submit handwritten solutions of the assignment to your instructor in tutorial
class (i.e., on 16/11/2007, Friday at 8.0 am) by mentioning y
ii
STOCHASTIC PROGRAMMING
11.1 INTRODUCTION
Stochastic or probabilistic programming deals with situations where some or
all of the parameters of the optimization problem are described by stochastic
(or random or probabilistic) variables rather than by det
8
GEOMETRIC PROGRAMMING
8.1 INTRODUCTION
Geometric programming is a relatively new method of solving a class of nonlinear programming problems. It was developed by Duffin, Peterson, and
Zener [8.1]. It is used to minimize functions that are in the form of
APPENDIX A
CONVEX AND CONCAVE FUNCTIONS
Convex Function A function /(X) is said to be convex if for any pair of
points
Xj
Aj
r0)
X1 =
r(2)
Lj
1
j
and X 2 =
.
J
L ? J
and all X, 0 < X < 1,
/[XX2 + (1 - X) X1] < X/(X2) + (1 - X)Z(X1)
(A.I)
that is, if the s
9
DYNAMIC PROGRAMMING
9.1 INTRODUCTION
In most practical problems, decisions have to be made sequentially at different
points in time, at different points in space, and at different levels, say, for a
component, for a subsystem, and/or for a system. The p
1
INTRODUCTION TO OPTIMIZATION
1.1 INTRODUCTION
Optimization is the act of obtaining the best result under given circumstances.
In design, construction, and maintenance of any engineering system, engineers
have to take many technological and managerial de
4
LINEAR PROGRAMMING II:
ADDITIONAL TOPICS AND
EXTENSIONS
4.1 INTRODUCTION
If a LP problem involving several variables and constraints is to be solved by
using the simplex method described in Chapter 3, it requires a large amount
of computer storage and t
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The direct compressive stress (ac) in the column due to the weight of the water
tank is given by
Mg
Mg
bd
XxX2
and the buckling stress for a fixed-free column (ab) is given by [1.71]
_ /VISA _T?EX\
a
bd ~ 48/2
~ yw)
(Efi)
To avoid failure o
COURSE: OPTIMIZATION TECHNIQUE AAOC C222
ASSIGNMENT - 6
Date of Assignment: 10/11/09
Date of Submission: 27/11/09 (Common Hour)
Maximum marks: 5
Q1. Solve the following by Branch and Bound Algorithm.
Max z = 3x1 4 x 2
s.t.
3x1 x 2 12, 3x1 + 11x 2 66,
x1 ,
COURSE: OPTIMIZATION TECHNIQUE AAOC C222
ASSIGNMENT-2
Date of Assignment: 11/09/09 Date of Submission 18/8/09 (Common Hour)
Maximum marks: 5
1(a) Why artificial variables are not called slack?
1(b) What is the main objective of Phase-I of the Two Phase Me
Determination of
Starting Basic Feasible
Solution
Determination of the starting Solution
In any transportation model we determine a starting
BFS and then iteratively move towards the optimal
solution which has the least shipping cost.
There are three meth
Explanation of the
entries in any simplex
tableau in terms of the
entries of the starting
tableau
In this lecture we explain how the
starting Simplex tableau (in matrix
form) gets transformed after some
iterations. We also give the meaning of
the entries
Addition of a new constraint
The addition of a new constraint to an existing
model can lead to one of two cases:
1. The new constraint is redundant, meaning
that it is satisfied by the current optimal
solution and hence can be dropped
altogether from the
Problem 6 Problem Set 2.3A Page
26(Modified)
Electra produces two types of electric motors,
each on a separate assembly line. The respective
daily capacities of the two lines are 150 and 200
motors. Type I motor uses 2 units of a certain
electronic compon
Problem 7 Problem Set 8.1A Page 351
Two products are manufactured on two sequential
machines. The following table gives the machining
times in minutes per unit for the two products:
Machine
1
2
Machining Time in min
Product 1
Product2
5
3
6
2
The daily pr